1.9 inverse functions - central bucks school district · inverse functions can be used to model and...
TRANSCRIPT
What you should learn• Find inverse functions infor-
mally and verify that two functions are inverse functionsof each other.
• Use graphs of functions todetermine whether functionshave inverse functions.
• Use the Horizontal Line Test to determine if functions areone-to-one.
• Find inverse functions alge-braically.
Why you should learn itInverse functions can be used to model and solve real-lifeproblems. For instance, inExercise 80 on page 101, aninverse function can be used to determine the year in whichthere was a given dollar amountof sales of digital cameras in theUnited States.
© Tim Boyle/Getty Images
Section 1.9 Inverse Functions 93
Inverse Functions1.9
Inverse FunctionsRecall from Section 1.4, that a function can be represented by a set of orderedpairs. For instance, the function from the set to theset can be written as follows.
In this case, by interchanging the first and second coordinates of each of theseordered pairs, you can form the inverse function of which is denoted by It is a function from the set to the set and can be written as follows.
Note that the domain of is equal to the range of and vice versa, as shown inFigure 1.92. Also note that the functions and have the effect of “undoing”each other. In other words, when you form the composition of with or thecomposition of with you obtain the identity function.
FIGURE 1.92
Finding Inverse Functions Informally
Find the inverse function of Then verify that both andare equal to the identity function.
SolutionThe function multiplies each input by 4. To “undo” this function, you need todivide each input by 4. So, the inverse function of is
You can verify that both and as follows.
Now try Exercise 1.
f �1� f �x�� � f �1�4x� �4x
4� xf � f �1�x�� � f �x
4� � 4�x
4� � x
f �1� f �x�� � xf� f �1�x�� � x
f �1�x� �x
4.
f �x� � 4xf
f �1� f �x��f � f �1�x��f(x) � 4x.
f (x) = x + 4
f −1(x) = x − 4
Domain of f
Range of f −1
Range of f
Domain of f −1
x f (x)
f �1� f �x�� � f �1�x � 4� � �x � 4� � 4 � x
f� f �1�x�� � f �x � 4� � �x � 4� � 4 � x
f,f �1f �1f
f �1ff �1,f
��5, 1�, �6, 2�, �7, 3�, �8, 4��f �1�x� � x � 4:
A,Bf �1.f,
��1, 5�, �2, 6�, �3, 7�, �4, 8��f�x� � x � 4:
B � �5, 6, 7, 8�A � �1, 2, 3, 4�f �x� � x � 4
Example 1
333202_0109.qxd 12/7/05 8:45 AM Page 93
94 Chapter 1 Functions and Their Graphs
Don’t be confused by the use of to denote the inverse function In thistext, whenever is written, it always refers to the inverse function of the func-tion and not to the reciprocal of
If the function is the inverse function of the function it must also be truethat the function is the inverse function of the function For this reason, youcan say that the functions and are inverse functions of each other.
Verifying Inverse Functions
Which of the functions is the inverse function of
SolutionBy forming the composition of with you have
Substitute for
Because this composition is not equal to the identity function it follows that is not the inverse function of By forming the composition of with you have
So, it appears that is the inverse function of You can confirm this by showingthat the composition of with is also equal to the identity function.
Now try Exercise 5.
fhf.h
f �h�x�� � f �5
x� 2� �
5
�5
x� 2� � 2
�5
�5
x�� x.
h,ff.gx,
� x. �25
x � 12
x.x � 2
5 �5
�x � 25 � � 2
f �g�x�� � f �x � 25 �
g,f
h�x� �5
x� 2g�x� �
x � 2
5
f�x� �5
x � 2?
gfg.f
f,gf �x�.f
f �1f �1.�1
Definition of Inverse FunctionLet and be two functions such that
for every in the domain of
and
for every in the domain of
Under these conditions, the function is the inverse function of the func-tion The function is denoted by (read “ -inverse”). So,
and
The domain of must be equal to the range of and the range of mustbe equal to the domain of f �1.
ff �1,f
f �1� f �x�� � x.f � f �1�x�� � x
ff �1gf.g
f.xg� f �x�� � x
gxf �g�x�� � x
gf
Additional Examples
Determine whether the two functionsare inverse functions of each other.
a.
b.
c.
Solution
The functions in parts (b) and (c) areinverse functions of each other because
and The functions in part (a) are not inversefunctions of each other.
g�f �x�� � x.f �g�x�� � x
f �x� � 2x � 4, g�x� �12
x � 2
f �x� �14
x � 3, g�x� � 12 � 4x
f �x� � 3x � 2, g�x� �13
x � 2
Consider the functions given by
and
Evaluate andfor the indicated
values of What can youconclude about the functions?
x.f�1� f �x��
f � f�1�x��
f�1�x� � x � 2.
x � 2f �x� �
Exploration
Example 2
x 0 7 45
f�1� f �x��
f � f�1�x��
�10
333202_0109.qxd 12/7/05 8:45 AM Page 94
Section 1.9 Inverse Functions 95
The Graph of an Inverse FunctionThe graphs of a function and its inverse function are related to each otherin the following way. If the point lies on the graph of then the point must lie on the graph of and vice versa. This means that the graph of isa reflection of the graph of in the line as shown in Figure 1.93.
Finding Inverse Functions Graphically
Sketch the graphs of the inverse functions and on the same rectangular coordinate system and show that the graphs are reflec-tions of each other in the line
SolutionThe graphs of and are shown in Figure 1.94. It appears that the graphs arereflections of each other in the line You can further verify this reflectiveproperty by testing a few points on each graph. Note in the following list that ifthe point is on the graph of the point is on the graph of
Graph of Graph of
Now try Exercise 15.
Finding Inverse Functions Graphically
Sketch the graphs of the inverse functions and onthe same rectangular coordinate system and show that the graphs are reflectionsof each other in the line
SolutionThe graphs of and are shown in Figure 1.95. It appears that the graphs arereflections of each other in the line You can further verify this reflectiveproperty by testing a few points on each graph. Note in the following list that ifthe point is on the graph of the point is on the graph of
Graph of Graph of
Try showing that and
Now try Exercise 17.
f �1� f �x�� � x.f � f �1�x�� � x
�9, 3��3, 9�
�4, 2��2, 4�
�1, 1��1, 1�
�0, 0��0, 0�f �1�x� � �xx ≥ 0f�x� � x2,
f �1.�b, a�f,�a, b�
y � x.f �1f
y � x.
f �1�x� � �x�x ≥ 0�f �x� � x2
�3, 3��3, 3�
�1, 2��2, 1�
��1, 1��1, �1�
��3, 0��0, �3�
��5, �1���1, �5�f �1�x� �
12�x � 3�f�x� � 2x � 3
f �1.�b, a�f,�a, b�
y � x.f �1f
y � x.
f �1�x� �12�x � 3�f �x� � 2x � 3
y � x,ff �1f �1,�b, a�f,�a, b�
f �1f
x
(a, b)
(b, a)
y = f (x)
y = x
y = f −1(x)
y
FIGURE 1.93
x
( 5, 1)− −
( 1, 5)− −
(0, 3)−
(1, 1)−
(2, 1)(3, 3)
(1, 2)
( 3, 0)−( 1, 1)−
−6 6
6
y x=
f x x−1( ) = ( + 3)12 f x x( ) = 2 3−
y
FIGURE 1.94
x
(3, 9)
(2, 4)
(4, 2)(9, 3)
(1, 1)
(0, 0) 3 4 5 6 7 8 9
1
2
3
4
5
6
7
8
9
f (x) = x2
y = x
f (x) = x−1
y
FIGURE 1.95
Example 3
Example 4
333202_0109.qxd 12/7/05 8:45 AM Page 95
96 Chapter 1 Functions and Their Graphs
One-to-One FunctionsThe reflective property of the graphs of inverse functions gives you a nicegeometric test for determining whether a function has an inverse function. Thistest is called the Horizontal Line Test for inverse functions.
If no horizontal line intersects the graph of at more than one point, then no-value is matched with more than one -value. This is the essential characteristic
of what are called one-to-one functions.
Consider the function given by The table on the left is a table ofvalues for The table of values on the right is made up by interchang-ing the columns of the first table. The table on the right does not represent a func-tion because the input is matched with two different outputs: and
So, is not one-to-one and does not have an inverse function.
Applying the Horizontal Line Test
a. The graph of the function given by is shown in Figure 1.96.Because no horizontal line intersects the graph of at more than one point,you can conclude that is a one-to-one function and does have an inversefunction.
b. The graph of the function given by is shown in Figure 1.97.Because it is possible to find a horizontal line that intersects the graph of atmore than one point, you can conclude that is not a one-to-one function anddoes not have an inverse function.
Now try Exercise 29.
ff
f�x� � x2 � 1
ff
f �x� � x3 � 1
f �x� � x2y � 2.y � �2x � 4
f �x� � x2.f �x� � x2.
xyf
Horizontal Line Test for Inverse FunctionsA function has an inverse function if and only if no horizontal lineintersects the graph of at more than one point.f
f
One-to-One FunctionsA function is one-to-one if each value of the dependent variable corre-sponds to exactly one value of the independent variable. A function has aninverse function if and only if is one-to-one.f
ff
x−2 −1−3 2 3
3
1
−3
−2 f (x) = x3 − 1
y
FIGURE 1.96
x−2−3 2 3
3
2
−3
−2 f (x) = x2 − 1
y
FIGURE 1.97
x y
4
1
0 0
1 1
4 2
9 3
�1
�2
Example 5
x
4
1
0 0
1 1
2 4
3 9
�1
�2
f �x� � x2
333202_0109.qxd 12/7/05 8:46 AM Page 96
Section 1.9 Inverse Functions 97
Finding Inverse Functions AlgebraicallyFor simple functions (such as the one in Example 1), you can find inverse func-tions by inspection. For more complicated functions, however, it is best to use thefollowing guidelines. The key step in these guidelines is Step 3—interchangingthe roles of and This step corresponds to the fact that inverse functions haveordered pairs with the coordinates reversed.
Finding an Inverse Function Algebraically
Find the inverse function of
SolutionThe graph of is a line, as shown in Figure 1.98. This graph passes the HorizontalLine Test. So, you know that is one-to-one and has an inverse function.
Write original function.
Replace by
Interchange and
Multiply each side by 2.
Isolate the -term.
Solve for
Replace by
Note that both and have domains and ranges that consist of the entire set ofreal numbers. Check that and
Now try Exercise 55.
f �1� f �x�� � x.f � f �1�x�� � xf �1f
f �1�x�.y f �1�x� �5 � 2x
3
y. y �5 � 2x
3
y 3y � 5 � 2x
2x � 5 � 3y
y.x x �5 � 3y
2
y.f �x� y �5 � 3x
2
f �x� �5 � 3x
2
ff
f �x� �5 � 3x
2.
y.x
Finding an Inverse Function1. Use the Horizontal Line Test to decide whether has an inverse function.
2. In the equation for replace by
3. Interchange the roles of and and solve for
4. Replace by in the new equation.
5. Verify that and are inverse functions of each other by showing thatthe domain of is equal to the range of the range of is equal to thedomain of and f � f �1�x�� � x and f �1� f �x�� � x.f �1,
ff �1,ff �1f
f �1�x�y
y.y,x
y.f �x�f �x�,
f
x−4−6
− 4
−2
4
4
6
6
−6
5 − 3xf x( ) =2
−2
y
FIGURE 1.98
Note what happens when you tryto find the inverse function of afunction that is not one-to-one.
You obtain two -values foreach x.
y
y � ±�x � 1
x � 1 � y2
x � y2 � 1
y � x2 � 1
f �x� � x2 � 1Originalfunction
Replace f(x) by y.
Interchange xand y.
Isolatey-term.
Solve for y.
Restrict the domain ofto Use
a graphing utility to graph thefunction. Does the restrictedfunction have an inverse func-tion? Explain.
x ≥ 0.f �x� � x2 � 1
Exploration
Example 6
333202_0109.qxd 12/7/05 8:46 AM Page 97
98 Chapter 1 Functions and Their Graphs
Finding an Inverse Function
Find the inverse function of
SolutionThe graph of is a curve, as shown in Figure 1.99. Because this graph passes theHorizontal Line Test, you know that is one-to-one and has an inverse function.
Write original function.
Replace by
Interchange and
Cube each side.
Solve for
Replace by
Both and have domains and ranges that consist of the entire set of realnumbers. You can verify this result numerically as shown in the tables below.
Now try Exercise 61.
f�1f
f �1�x�.y x3 � 1 � f �1�x�
y. x3 � 1 � y
x3 � y � 1
y.x x � 3�y � 1
y.f �x� y � 3�x � 1
f �x� � 3�x � 1
ff
f �x� � 3�x � 1.
Activities
1. Given find
Answer:
2. Show that and are inverse func-tions of each other by showing that
and
3. Describe the graphs of functions thathave inverse functions and show howthe graphs of a function and itsinverse function are related.
g�x� � 3�x � 13
f �x� � 3x 3 � 1
g�f �x�� � x.f �g�x�� � x
gf
f �1�x� �x � 7
5
f �1�x�.f �x� � 5x � 7,
x−3
−2
1 3
−3
f x x( ) = + 1
−2
3
2
3
−1
2
y
FIGURE 1.99
x
0
0 1
7 2
26 3
�1
�1�2
�2�9
�3�28
f �x� x
0
1 0
2 7
3 26
�1
�2�1
�9�2
�28�3
f �1�x�
W RITING ABOUT MATHEMATICS
The Existence of an Inverse Function Write a short paragraphdescribing why the following functions do or do not haveinverse functions.
a. Let represent the retail price of an item (in dollars), andlet represent the sales tax on the item. Assume thatthe sales tax is 6% of the retail price and that the sales taxis rounded to the nearest cent. Does this function have an inverse function? (Hint: Can you undo this function?
For instance, if you know that the sales tax is $0.12, canyou determine exactly what the retail price is?)
b. Let represent the temperature in degrees Celsius, and let represent the temperature in degrees Fahrenheit.Does this function have an inverse function? (Hint: The formula for converting from degrees Celsius to degreesFahrenheit is )F �
95 C � 32.
f �x�x
f �x�x
Example 7
333202_0109.qxd 12/7/05 8:46 AM Page 98
Section 1.9 Inverse Functions 99
Exercises 1.9
In Exercises 1– 8, find the inverse function of informally.Verify that and
1. 2.
3. 4.
5. 6.
7. 8.
In Exercises 9–12, match the graph of the function with thegraph of its inverse function. [The graphs of the inversefunctions are labeled (a), (b), (c), and (d).]
(a) (b)
(c) (d)
9. 10.
11. 12.
In Exercises 13–24, show that and are inverse functions(a) algebraically and (b) graphically.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24. g�x� �2x � 3x � 1
f �x� �x � 3x � 2
,
g�x� � �5x � 1x � 1
f �x� �x � 1x � 5
,
0 < x ≤ 1g�x� �1 � x
x,x ≥ 0,f �x� �
1
1 � x,
x ≤ 9g�x� � �9 � x,x ≥ 0,f �x� � 9 � x2,
g�x� � 3�1 � xf �x� � 1 � x3,
x ≥ 0g�x� � x2 � 4,f �x� � �x � 4,
g�x� �1
xf �x� �
1
x,
g�x� � 3�8xf �x� �x3
8,
g�x� �3 � x
4f �x� � 3 � 4x,
g�x� �x � 1
7f �x� � 7x � 1,
g�x� � x � 5f �x� � x � 5,
g�x� �x
2f �x� � 2x,
gf
x2 31
2
1
−2−3
−3
3
y
x42 31
2
4
1
3
y
x4 5 62 31
21
3456
y
x2 3 41−1−2
2
1
3
4
y
x2 31
2
−2
−2
−3
−3
1
3
y
x2 31−1
2
−2
1
3
4
y
x4 5 62 31
21
3456
y
x42 31
2
4
1
3
y
f �x� � x5f �x� � 3�x
f �x� �x � 1
5f �x� � 3x � 1
f �x� � x � 4f �x� � x � 9
f �x� �13 xf �x� � 6x
f �1�f �x � x.f �f �1�x � xf
VOCABULARY CHECK: Fill in the blanks.
1. If the composite functions and then the function is the ________ function of
2. The domain of is the ________ of and the ________ of is the range of
3. The graphs of and are reflections of each other in the line ________.
4. A function is ________ if each value of the dependent variable corresponds to exactly one value of the independent variable.
5. A graphical test for the existence of an inverse function of is called the _______ Line Test.
PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com.
f
f
f�1f
f.f�1f�1,f
f.gg� f�x�� � xf�g�x�� � x
333202_0109.qxd 12/7/05 8:46 AM Page 99
100 Chapter 1 Functions and Their Graphs
In Exercises 25 and 26, does the function have an inversefunction?
25.
26.
In Exercises 27 and 28, use the table of values for to complete a table for
27.
28.
In Exercises 29–32, does the function have an inversefunction?
29. 30.
31. 32.
In Exercises 33–38, use a graphing utility to graph the func-tion, and use the Horizontal Line Test to determine whetherthe function is one-to-one and so has an inverse function.
33. 34.
35.
36.
37.
38.
In Exercises 39–54, (a) find the inverse function of (b) graph both and on the same set of coordinate axes,(c) describe the relationship between the graphs of and
, and (d) state the domain and range of and .
39. 40.
41. 42.
43. 44.
45.
46.
47. 48.
49. 50.
51. 52.
53. 54.
In Exercises 55–68, determine whether the function has aninverse function. If it does, find the inverse function.
55. 56.
57. 58.
59. 60.
61. 62.
63. 64.
65. 66.
67. 68.
x4
2
1
1
4
−1
−2
−2
3
2 3
y
x2
2
1
1
4
−1
−2
−2−3−4
3
y
f �x� � �x � 2f �x� � �2x � 3
x2
2
1 3
1
4
4 5 6−1−2−3−4
y
x
y
1
1
−2
−1
x ≤ 2f �x� �x � 2,h�x� � �4
x2
f �x� � ��x,x2 � 3x,
x ≤ 0 x > 0
f �x� � �x � 3,6 � x,
x < 0 x ≥ 0
q�x� � �x � 5�2x ≥ �3f �x� � �x � 3�2,
f �x� �3x � 4
5p�x� � �4
f �x� � 3x � 5g�x� �x
8
f �x� �1
x2f �x� � x4
f �x� �8x � 42x � 6
f �x� �6x � 44x � 5
f �x� � x3�5f �x� � 3�x � 1
f �x� �x � 3x � 2
f �x� �x � 1x � 2
f �x� � �2x
f �x� �4x
x ≤ 0f �x� � x2 � 2,
0 ≤ x ≤ 2f �x� � �4 � x2,
x ≥ 0f �x� � x2,f �x� � �x
f �x� � x3 � 1f �x� � x5 � 2
f �x� � 3x � 1f �x� � 2x � 3
f �1ff �1
ff �1f
f,
f �x� �18�x � 2�2 � 1
f �x� � �2x�16 � x2
g�x� � �x � 5�3
h�x� � x � 4 � x � 4f �x� � 10g�x� �
4 � x
6
x2 4 6
−2
2
4
y
x2
−2
−2
2
y
x2 4−2−4
−2
2
6
y
x2 4 6
−2
2
6
4
y
y � f �1�x.y � f �x
x 0 1 2 3
0 2 4 6 8�2f �x�
�1�2
x 0 1 2
2 5�1�4�7�10f �x�
�1�2�3
x 0 1 2 3 4
1 2 1 �6�2�2f �x�
�1
x 0 2 3
10 6 4 1 �10�3f �x�
�1�2�3
333202_0109.qxd 12/7/05 8:46 AM Page 100
Section 1.9 Inverse Functions 101
In Exercises 69–74, use the functions given byand to find the indicated value or
function.
69. 70.
71. 72.
73. 74.
In Exercises 75–78, use the functions given by and to find the specified function.
75. 76.
77. 78.
80. Digital Camera Sales The factory sales (in millions ofdollars) of digital cameras in the United States from 1998through 2003 are shown in the table. The time (in years) isgiven by with corresponding to 1998. (Source:Consumer Electronincs Association)
(a) Does exist?
(b) If exists, what does it represent in the context of theproblem?
(c) If exists, find
(d) If the table was extended to 2004 and if the factorysales of digital cameras for that year was $2794 million, would exist? Explain.
81. Miles Traveled The total numbers (in billions) of miles traveled by motor vehicles in the United Statesfrom 1995 through 2002 are shown in the table. The time(in years) is given by with corresponding to 1995.(Source: U.S. Federal Highway Administration)
(a) Does exist?
(b) If exists, what does it mean in the context of theproblem?
(c) If exists, find
(d) If the table was extended to 2003 and if the totalnumber of miles traveled by motor vehicles for thatyear was 2747 billion, would exist? Explain.f�1
f�1�2632�.f�1
f�1
f�1
t � 5t,
f
f�1
f�1�1825�.f�1
f�1
f�1
t � 8t,
f
�g � f ��1� f � g��1
f �1 � g�1g�1� f
�1
g�x � 2x � 5f �x � x � 4
g�1 � f �1( f � g)�1
�g�1 �g�1���4�� f �1 � f �1��6��g�1 � f �1���3�� f �1 � g�1��1�
g�x � x 3 f �x � 18 x � 3
Year, t Sales,
8 519
9 1209
10 1825
11 1972
12 2794
13 3421
f �t�
79. U.S. Households The numbers of households (inthousands) in the United States from 1995 to 2003 areshown in the table. The time (in years) is given by with corresponding to 1995. (Source: U.S.Census Bureau)
(a) Find
(b) What does mean in the context of the problem?
(c) Use the regression feature of a graphing utility tofind a linear model for the data,(Round and to two decimal places.)
(d) Algebraically find the inverse function of the linearmodel in part (c).
(e) Use the inverse function of the linear model youfound in part (d) to approximate
(f) Use the inverse function of the linear model youfound in part (d) to approximate Howdoes this value compare with the original datashown in the table?
f�1�108,209�.
f�1�117, 022�.
bmy � mx � b.
f�1
f�1�108,209�.
t � 5t,
f
Model It
Year, t Households,
5 98,990
6 99,627
7 101,018
8 102,528
9 103,874
10 104,705
11 108,209
12 109,297
13 111,278
f �t
Year, t Miles traveled,
5 2423
6 2486
7 2562
8 2632
9 2691
10 2747
11 2797
12 2856
f �t
333202_0109.qxd 12/7/05 8:46 AM Page 101
102 Chapter 1 Functions and Their Graphs
82. Hourly Wage Your wage is $8.00 per hour plus $0.75 foreach unit produced per hour. So, your hourly wage interms of the number of units produced is
(a) Find the inverse function.
(b) What does each variable represent in the inversefunction?
(c) Determine the number of units produced when yourhourly wage is $22.25.
83. Diesel Mechanics The function given by
approximates the exhaust temperature in degreesFahrenheit, where is the percent load for a diesel engine.
(a) Find the inverse function. What does each variablerepresent in the inverse function?
(b) Use a graphing utility to graph the inverse function.
(c) The exhaust temperature of the engine must not exceed500 degrees Fahrenheit. What is the percent loadinterval?
84. Cost You need a total of 50 pounds of two types ofground beef costing $1.25 and $1.60 per pound,respectively. A model for the total cost of the two typesof beef is
where is the number of pounds of the less expensiveground beef.
(a) Find the inverse function of the cost function. Whatdoes each variable represent in the inverse function?
(b) Use the context of the problem to determine thedomain of the inverse function.
(c) Determine the number of pounds of the less expensiveground beef purchased when the total cost is $73.
Synthesis
True or False? In Exercises 85 and 86, determine whetherthe statement is true or false. Justify your answer.
85. If is an even function, exists.
86. If the inverse function of exists and the graph of has a-intercept, the -intercept of is an -intercept of
87. Proof Prove that if and are one-to-one functions, then
88. Proof Prove that if is a one-to-one odd function, thenis an odd function.
In Exercises 89– 92, use the graph of the function to createa table of values for the given points. Then create a secondtable that can be used to find , and sketch the graph of
if possible.
89. 90.
91. 92.
93. Think About It The function given by
has an inverse function, and Find
94. Think About It The function given by
has an inverse function, and Find
Skills Review
In Exercises 95–102, solve the equation using anyconvenient method.
95.
96.
97.
98.
99.
100.
101.
102.
103. Find two consecutive positive even integers whoseproduct is 288.
104. Geometry A triangular sign has a height that is twice itsbase. The area of the sign is 10 square feet. Find the baseand height of the sign.
2x2 � 4x � 9 � 2�x � 1�2
50 � 5x � 3x2
2x2 � 4x � 6 � 0
x2 � 6x � 4 � 0
9x2 � 12x � 3 � 0
4x2 � 12x � 9 � 0
�x � 5�2 � 8
x2 � 64
k.f �1(�5) � 2.
f �x� � k�x3 � 3x � 4�
k.f �1(3) � �2.
f �x� � k�2 � x � x3�
x
−2−2−4 4
f
−4
y
x
−4
−2
4
6
f
4
y
x−2 2
2
4
f
−4
4
y
x4
f4
2 86
2
6
8
y
f �1
f �1
f
f�1f
� f � g��1�x� � �g�1� f�1��x�.
gf
f �1.xfyyff
f �1f
x
y � 1.25x � 1.60�50 � x�
y
xy
0 < x < 100y � 0.03x2 � 245.50,
y � 8 � 0.75x.
y
333202_0109.qxd 12/7/05 8:46 AM Page 102